Note that the case <math>p = 2</math>, where the group becomes [[dihedral group:D8]], behaves somewhat differently from the general case. We note on the page all the places where the discussion does not apply to <math>p = 2</math>.

These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics and so the group is sometimes called a finite Heisenberg group. Generators <math>x,y,z</math> correspond to matrices:

These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics and so the group is sometimes called a finite Heisenberg group. Generators <math>x,y,z</math> correspond to matrices:

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0 & 0 & 1\\

0 & 0 & 1\\

\end{pmatrix}</math>

\end{pmatrix}</math>

+

+

Note that in the above presentation, the generator <math>z</math> is redundant, and the presentation can thus be rewritten as a presentation with only two generators <math>x</math> and <math>y</math>.

===As a semidirect product===

===As a semidirect product===

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# These groups fall in the more general family <math>UT(n,p)</math> of [[unitriangular matrix group]]s. The unitriangular matrix group <math>UT(n,p)</math> can be described as the group of unipotent upper-triangular matrices in <math>GL(n,p)</math>, which is also a <math>p</math>-Sylow subgroup of the [[general linear group]] <math>GL(n,p)</math>. This further can be generalized to <math>UT(n,q)</math> where <math>q</math> is the power of a prime <math>p</math>. <math>UT(n,q)</math> is the <math>p</math>-Sylow subgroup of <math>GL(n,q)</math>.

# These groups fall in the more general family <math>UT(n,p)</math> of [[unitriangular matrix group]]s. The unitriangular matrix group <math>UT(n,p)</math> can be described as the group of unipotent upper-triangular matrices in <math>GL(n,p)</math>, which is also a <math>p</math>-Sylow subgroup of the [[general linear group]] <math>GL(n,p)</math>. This further can be generalized to <math>UT(n,q)</math> where <math>q</math> is the power of a prime <math>p</math>. <math>UT(n,q)</math> is the <math>p</math>-Sylow subgroup of <math>GL(n,q)</math>.

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# These groups also fall into the general family of [[extraspecial group]]s.

+

# These groups also fall into the general family of [[extraspecial group]]s. For any number of the form <math>p^{1 + 2m}</math>, there are two extraspecial groups of that order: an extraspecial group of "+" type and an extraspecial group of "-" type. <math>UT(3,p)</math> is an extraspecial group of order <math>p^3</math> and "+" type. The other type of extraspecial group of order <math>p^3</math>, i.e., the extraspecial group of order <math>p^3</math> and "-" type, is [[semidirect product of cyclic group of prime-square order and cyclic group of prime order]].

| [[Inner automorphism group]] || [[Inner automorphism group::Elementary abelian group of prime-square order]] || It is the quotient by the center, which is of prime order.

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|-

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| [[Abelianization]] || [[Abelianization::Elementary abelian group of prime-square order]] || It is the quotient by the commutator subgroup, which is of prime order.

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|-

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| [[Frattini quotient]] || [[Frattini quotient::Elementary abelian group of prime-square order]] || It is the quotient by the Frattini subgroup, which is of prime order.

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|}

==Endomorphisms==

==Endomorphisms==

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The automorphisms essentially permute the subgroups of order <math>p^2</math> containing the center, while leaving the center itself unmoved.

The automorphisms essentially permute the subgroups of order <math>p^2</math> containing the center, while leaving the center itself unmoved.

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==Related groups==

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For any prime <math>p</math>, there are (up to isomorphism) two non-abelian groups of order <math>p^3</math>. One of them is this, and the [[prime-cube order group:p2byp|other]] is the semidirect product of the cyclic group of order <math>p^2</math> by a group of order <math>p</math> acting by power maps (with the generator corresponding to multiplication by <math>p+1</math>).

Definition by presentation

These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics and so the group is sometimes called a finite Heisenberg group. Generators correspond to matrices:

Note that in the above presentation, the generator is redundant, and the presentation can thus be rewritten as a presentation with only two generators and .

As a semidirect product

This group of order can also be described as a semidirect product of the elementary abelian group of order by the cyclic group of order, with the following action. Denote the base of the semidirect product as ordered pairs of elements from . The action of the generator of the acting group is as follows:

In this case, for instance, we can take the subgroup with as the elementary abelian subgroup of order , i.e., the elementary abelian subgroup of order is the subgroup:

The acting subgroup of order can be taken as the subgroup with , i.e., the subgroup:

Families

These groups fall in the more general family of unitriangular matrix groups. The unitriangular matrix group can be described as the group of unipotent upper-triangular matrices in , which is also a -Sylow subgroup of the general linear group. This further can be generalized to where is the power of a prime . is the -Sylow subgroup of .